MNRAS 461, 4406–4418 (2016) doi:10.1093/mnras/stw1577 Advance Access publication 2016 July 4
On the orbital evolution of a pair of giant planets in mean motion resonance
Q. Andre´1,2‹ and J. C. B. Papaloizou1 1DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 OWA, UK 2Departement´ de Physique, ENS Cachan, Universite´ Paris-Saclay, 61 Avenue du President´ Wilson, F-94230 Cachan, France
Accepted 2016 June 30. Received 2016 June 30; in original form 2015 July 9
ABSTRACT Pairs of extrasolar giant planets in a mean motion commensurability are common with 2:1 Downloaded from resonance occurring most frequently. Disc–planet interaction provides a mechanism for their origin. However, the time-scale on which this could operate in particular cases is unclear. We perform 2D and 3D numerical simulations of pairs of giant planets in a protoplanetary disc as they form and maintain a mean motion commensurability. We consider systems with http://mnras.oxfordjournals.org/ current parameters similar to those of HD 155358, 24 Sextantis and HD 60532, and disc models of varying mass, decreasing mass corresponding to increasing age. For the lowest mass discs, systems with planets in the Jovian mass range migrate inwards maintaining a 2:1 commensurability. Systems with the inner planet currently at around 1 au from the central star could have originated at a few au and migrated inwards on a time-scale comparable to protoplanetary disc lifetimes. Systems of larger mass planets such as HD 60532 attain 3:1 resonance as observed. For a given mass accretion rate, results are insensitive to the disc model for the range of viscosity prescriptions adopted, there being good agreement between 2D and 3D simulations. However, in a higher mass disc a pair of Jovian mass planets passes through at University of Cambridge on August 10, 2016 2:1 resonance before attaining a temporary phase lasting a few thousand orbits in an unstable 5:3 resonance prior to undergoing a scattering. Thus, finding systems in this commensurability is unlikely. Key words: accretion, accretion discs – hydrodynamics – methods: numerical – planet–disc interactions – protoplanetary discs – planetary systems.
figurations in situ is expected to be small (e.g. Beauge, Ferraz-Mello 1 INTRODUCTION & Michtchenko 2012). Pairs of extrasolar giant planets in a mean motion commensurability Disc–planet interaction can produce the required evolution of the are a common occurrence. It has been estimated that sixth of multi- semi-major axes. This may result in convergent migration leading planet systems detected by the radial velocity technique are in or to the formation of a commensurability (see Baruteau et al. 2014 close to a 2:1 commensurability (Wright et al. 2011). Parameters for and references therein). Accordingly, understanding the observed some cases of interest that are considered in this paper are shown configuration of such systems has the potential for either revealing in Table 1 (for additional examples see e.g. Emelyanenko 2012). how disc–planet interactions may have operated or for ruling them In addition, there are two known systems in 3:2 resonance (Correia out. et al. 2009;Reinetal.2012; Robertson et al. 2012b) and one in a Previous numerical studies of commensurabilities forming and 4:3 resonance (Johnson et al. 2011;Reinetal.2012) consistent with evolving as a result of disc–planet interactions have focused on the systems in 2:1 resonance being the most commonly observed systems such as GJ 876, HD 45364 and HD 6805 interacting with commensurability. disc modelled with either constant kinematic viscosity or with the The existence of these resonant systems indicates that dissipative α-viscosity parameter of Shakura & Sunyaev (1973) taken to be con- mechanisms that result in changes to planet semi-major axes that stant (for a review see Baruteau et al. 2014 and references therein). produce related changes to period ratios in planetary systems have In this paper, we extend such studies, considering systems with or- operated. This is because the probability of forming resonant con- bital parameters similar to those of HD 155358 (Robertson et al. 2012a), 24 Sextantis (Johnson et al. 2011), HD 60532 (Laskar & Correia 2009) as well as HD 6805 (Trifonov et al. 2014) each E-mail: [email protected] of which have the inner planet with semi-major axis in the range